These days, you pretty much can't go wrong with any of the leading major brands (Nikon, Sony, Canon). Given your budget, I'd suggest that you pick the previous generations of any of them. I'm a Canon guy, and have two Canon 7D's and also a Sony NEX-7 (for travel and walk-around shooting). You can pick up a brand new Canon T4i (current generation of consumer DSLR) for < $600 on eBay (I recently bought one for my son)7. I bought a used 7D body for $1000 on CraigsList. There are lots of options. You can spend a LOT of time trying to identify the "perfect" camera, or you can get started.
Check sites like Digital Photography School if you want to read lots of reviews and get lots of tutorials.
I agree with the previous answer that the lens is at least as important as the camera body. That said, there are lots of excellent consumer-grade lenses (like the Tamron 18-270mm telephoto zoom or the Canon 18-200mm telephoto zoom) that are readily available, both new and used, and would provide you with excellent results as you learn your craft.
My suggestion would be - based purely on my irrational bias toward Canon - either a brand new T4i and a couple of lenses, or a used 7D for around $1000 and then one decent consumer lens. If you search eBay and Craigslist, you'll find some excellent options for either.
The advantage of the T4i (or any of the new crop of cameras) is that they do a fantastic job of HD video. The 7D (and the equivalent Nikon and Sony cameras) also do a very good job of video, so the trade-offs are slight.
I can pretty much guarantee that no matter what you start with, you'll be wanting to upgrade relatively soon, as you learn more and increase your skill and expose your preferences. Start with something you can afford, that has the features you need (they ALL do), and build up from there.